Physics 2B Chapter 17 Notes - Conduction Spring 2018

Size: px

Start display at page:

Download "Physics 2B Chapter 17 Notes - Conduction Spring 2018"

Eric Page
6 years ago
Views:

1 Heat as a Fluid Scientists in the 1700 s had no idea what was going on. They were figuring things out for the first time and they didn t have much to work with because the scientific method was still relatively new. All knowledge at the time was treated with a justifiable dose of healthy skepticism. It was in this era that a few brave souls pioneered the investigation into the nature of heat. Through a lot of careful experimentation, they were able to determine that heat is a form of energy and that it can be transferred from one object to another, apparently without any work being done. (Keep in mind that usually when energy is transferred from one object to another, it is through application of a force... i.e. the force applied and the distance the object moves.) The exchange of heat was mysterious in this era, and so the best they could do was describe heat as some kind of fluid. Recall that when we covered fluids in Chapter 14, I started the topic by asking What is a fluid? The answer was anything that flows. So even though we think of fluids as physical stuff (i.e. liquids, gases, and even some solids) that move in a way that we describe as flowing, there was nothing in our answer to this question that required a fluid to be tangible stuff. Which means that technically, heat does qualify as a fluid, by our definition. The flow is not of matter (i.e. stuff with mass), but of energy. We define the rate of heat flow: Q H = the amount of heat (i.e. Q ) that flows per time. t The unit of H is simply the unit of heat divided by the unit of time. In standard units, this would be Joule / sec Watt Thermal Conduction through a Wall How fast does heat flow? To answer that question we need some information. Specifically, we need to know where the heat is flowing, or what the heat is flowing through (like water flows through a pipe.) To accomplish this, we make a simple definition of a wall : a homogenous object (i.e. made of one material) in the shape of a rectangular solid (i.e. with uniform cross-section area and uniform width.) We define the type of material from which the wall is made by its thermal conductivity, a measure of how well the material conducts heat. We use a k to represent conductivity. We also can define the two temperatures on either side of the wall : one side is warmer, with temperature T H, and the other side is colder, with temperature T C. The heat always flows from the hot side to the cold side (much as water always flows downhill.) Page 1 of 6

2 If the wall has cross section area A and thickness w, then the rate of heat flow is given by the equation: H ka T w = where T = T H - T C Units of the conductivity, k: since H is in Watts, T is in Kelvin and A and w are in m 2 and m, respectively, the SI unit of k must be W/m-K. These units have no real physical meaning; they exist only to make the units for the above calculation come out the same on both sides of the equation. High values of k, for metals which conduct heat very well, are in the 200 to 400 range (in SI units.) Low values of k, for materials that insulate well (e.g. fiberglass, foam, wood) are in the 0.01 to 1.00 range. Note that this equation includes: the rate of heat flow, H; the difference in temperature, T; and three properties of the wall, k A and w. We can now define the resistance of the wall in terms of these three properties: w R = resistance of a wall ka And we can then rewrite the conduction equation to make use of this new definition: T = RH general form of thermal conduction equation This form is useful for a few reasons. First, we have defined R for a wall, but below we will also define R for a cylindrical shell. In general, an expression for R can be determined for an object of any Page 2 of 6

3 geometry. The above equation then applies to any object of any geometry; you just have to have the appropriate expression for R. Second, this version of the conduction equation is simpler; it includes only three things: the temperature difference, which is responsible for pushing the heat through the wall; the resistance of the wall (or cylindrical shell, or whatever); and the rate at which heat flows through. Essentially, this expression is analogous to Newton s Second Law: the force pushes an object, its mass resists, and the result is the acceleration of the object. Instead of F = ma, here we have T = RH. Walls in Series We started the discussion of conduction with a homogenous wall (i.e. one material) because it is a simple situation that is easy to define. But a more practical situation is one where a wall is made of two or more different materials. For example, a wall of a building might be made of some kind of structural material (e.g. stucco) but then covered with a layer of insulating material (e.g. fiberglass) and the interior might be finished with an appropriate material (e.g. drywall... and some paint.) We can define each of these layers as an individual wall of one material, and create the complete wall by arranging these walls in series. We refer to this arrangement as in series because the heat that flows from one side to the other must pass through each layer in succession. Series refers to the nature of the heat flow: through one layer, then the next, and the next, etc. We can use the conduction equation for each of the individual walls, which we can refer to as Wall 1 and Wall 2. These individual walls each have their own resistance, R 1 and R 2, their own T, and their own H, the rate of heat flow through each wall. Note that in the diagram we have a new temperature, T M. This is the middle temperature, i.e. the temperature where the two walls meet. This new T M will allow us to write the T for each wall in terms of the individual temperatures. Page 3 of 6

4 For our walls in series we can write: Wall 1: T 1 = R 1 H 1 Wall 2: T 2 = R 2 H 2 And we can also write, for both walls combined (i.e. for the total wall ): Total wall: T total = R total H total Now we make two simple observations: 1. Since T 1 = T H - T M and T 2 = T M - T C, and T total = T H - T C, then T total = T 1 + T 2 2. The rate at which heat flows through the first wall must be the same at which heat flows through the second wall, which must, by definition, be the rate at when heat flows through the total wall. So: H total = H 1 = H 2 Now using the equations above, we can substitute RH for each T and: R total H total = R 1 H 1 + R 2 H 2 Or: R total = R 1 + R 2 total resistance of walls in series For this derivation, we only used two walls in series. But we could have used any number and the result would have been the same: the total resistance of walls in series is simply the sum of the resistances of the individual walls. Resistance of a Cylindrical Shell Heat conduction through the walls of a cylindrical shell, i.e. a pipe, is of practical importance: many fluids that are transported through pipes must be maintained at a specific temperature and so the loss or gain of heat through the walls of the pipe must be taken into consideration when designing such a system. If the inside of a cylindrical shell is maintained at temperature T H and the outside is maintained at temperature T C, heat will flow through the shell in the same way that it did through our walls above. Therefore we can write, for the cylindrical shell: T = RH Page 4 of 6

5 just as we did for the wall. We now must figure out how to express the resistance of the cylindrical shell, in terms of its properties. We can define the inner and outer diameters of the shell as a and b, respectively, and we can define the length of the cylinder as L. We can then unfold the cylinder to create a flat surface of length L and width that varies from the inner circumference, πa, on one side, to the outer circumference, πb, on the other side: Since the area of this unrolled shell varies, we cannot use the expression for R that was defined above (i.e. the expression for a simple rectangular solid.) We can, however, slice the unrolled shell into very thin rectangular solids. Each of these thin slices will have width 2πr and length L, with thickness dr (i.e. an infinitely thin bit in the radial direction, as defined by the geometry of the cylinder.) We can use these definitions to write the resistance associated with one thin slice : dr dr = k 2πr L where dr takes the place of w and 2πrL is the area. As all of the thin slices are arranged in series, we can get the total resistance of the cylindrical shell by adding the resistance of all the slices: R = b a / 2 dr 1 = / 2 k 2πr L k 2 π L b / 2 a / 2 dr r Page 5 of 6

6 Or: R ln( b / a) = 2 π k L resistance of a cylindrical shell Note that for a situation in which two or more cylindrical shells are arranged in series (e.g. a layer of insulation surrounding a pipe), this expression can be used to determine the resistance of each individual shell, and then the resistances of the individual shells can be added to find the total resistance of the combination of shells. Page 6 of 6

Arctice Engineering Module 3a Page 1 of 32

Arctice Engineering Module 3a Page 1 of 32 Welcome back to the second part of the second learning module for Fundamentals of Arctic Engineering online. We re going to review in this module the fundamental principles of heat transfer. Exchange of